featurelesson geometry lesson main 1. if ab = 3x + 11, bc = 2x + 19, and cd = 7x – 17, find x. 2....
TRANSCRIPT
FeatureLesson
GeometryGeometry
LessonMain
1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x.
2. If m BAD = y and m ADC = 4y – 70, find y.
3. If m ABC = 2x + 100 and m ADC = 6x + 84, find m BCD.
4. If m BCD = 80 and m CAD = 34, find m ACD.
5. If AP = 3x, BP = y, CP = x + y, and DP = 6x – 40, find x and y.x = 10, y = 20
Use parallelogram ABCD for Exercises 1–5
7
50
72
46
Properties of ParallelogramsProperties of Parallelograms
Lesson 6-2
Lesson Quiz
6-3
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
(For help, go to Lessons 1-8 and 3-7.)
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Check Skills You’ll Need
Check Skills You’ll Need
1. Find the coordinates of the midpoints of AC and BD. What is the relationship between AC and BD?
2. Find the slopes of BC and AD. How do they compare?
3. Are AB and DC parallel? Explain.
4. What type of figure is ABCD?
Use the figure at the right for Exercises 1–4.
6-3
FeatureLesson
GeometryGeometry
LessonMain
1. For A(1, 2) and C(4, 1),
or (2.5, 1.5). For B(1, 0) and D(4, 3),
or (2.5, 1.5). They bisect one another.
1 + 42
2 + 12
x1 + x2
2
y1 + y2
2, = , = ,
52
32
x1 + x2
2
y1 + y2
21 + 4
20 + 3
2, =
,
= , 52
32
Solutions
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
.
Check Skills You’ll Need
6-3
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Solutions (continued)
Check Skills You’ll Need
2. For BC, the endpoints are B(1, 0) and C(4, 1);
For AD, the endpoints are A(1, 2) and D(4, 3);
The slopes of BC and AD are equal.
3. Yes; they are vertical lines.
4. From Exercise 2, the slopes of BC and AD are the same, so the lines are parallel. From Exercise 3, AB and DC are parallel. Thus ABCD is a parallelogram.
y2 – y1
x2 – x1
1 – 04 – 1
= =13m =
y2 – y1
x2 – x1
3 – 24 – 1
= = 13
m =
6-3
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
;WX ZY XY WZ
XZ XZ
WXZ YZX ;WXZ YZX WZX YXZ
& are AIAs;
are AIAs
WXZ YZX
WZX YXZ
;WX ZY XY WZ
Given
Reflexive POC
SSS
CPCTC
Defn of AIA
Converse of AIA Thm
Defn of WXYZ is a
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
6-3
Notes
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Notes
6-3
FeatureLesson
GeometryGeometry
LessonMain
Find values of x and y for which ABCD must be a parallelogram.
If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other.
If x = 18 and y = 89, then ABCD is a parallelogram.
10x – 24 = 8x + 12 Diagonals of parallelograms 2y – 80 = y + 9bisect each other.
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
x = 182x = 36
y = 89 Solve.
2x – 24 = 12 y – 80 = 9Collect the variable terms on one side.
Quick Check
Additional Examples
6-3
Finding Values for Parallelograms
FeatureLesson
GeometryGeometry
LessonMain
Determine whether the quadrilateral is a parallelogram. Explain.
a.
b.
a. All you know about the quadrilateral is that only one pair of opposite sides is congruent.
b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360.
Therefore, you cannot conclude that the quadrilateral is a parallelogram.
If x represents the measure of the unmarked angle, x + 75 + 105 + 75 = 360, so x = 105.
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-6. Quick Check
Additional Examples
6-3
Is the Quadrilateral a Parallelogram?
FeatureLesson
GeometryGeometry
LessonMain
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Quick Check
Additional Examples
6-3
The crossbars and the sections of the rulers are congruent no matter how they are positioned. So, ABCD is always a parallelogram. Since ABCD is a parallelogram, the rulers are parallel. Therefore, the direction the ship should travel is the same as the direction shown on the chart’s compass.
FeatureLesson
GeometryGeometry
LessonMain
The captain of a fishing boat plots a course toward a
school of bluefish. One side of a parallel rule connects the boat
with the school of bluefish. The other side makes a 36° angle
north of due east on the chart’s compass. Explain how the
captain knows in which direction to sail to reach the bluefish.
Because both sections of the rulers and the crossbars are congruent, the
rulers and crossbars form a parallelogram.
Therefore, the angle shown on the chart’s compass is congruent to the angle the boat should travel, which is 36° north of due east.
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Quick Check
Additional Examples
6-3
Real-World Connection
FeatureLesson
GeometryGeometry
LessonMain
Find the values of the variables for which GHIJ must be a parallelogram.
1. 2.
x = 6, y = 0.75 a = 34, b = 26
Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram
Lesson 6-3
Determine whether the quadrilateral must be a parallelogram. Explain.
3. 4. 5.
No; both pairs of opposite sides are not necessarily congruent.
Yes; the diagonals bisect each other.
Yes; one pair of opposite sides is both congruent and parallel.
Lesson Quiz
6-3